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EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2007; 36:24452453Published online 2 August 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.738

SHORT COMMUNICATION

Influence of the soilstructure interaction on the fundamentalperiod of buildings

Louay Khalil , Marwan Sadek,, and Isam Shahrour

Laboratoire de M ecanique de Lille (UMR 8107), Universit e des Sciences et Technologies de Lille,

Villeneuve dAscq, F-59655, France

SUMMARY

This paper includes an investigation of the influence of the soilstructure interaction (SSI) on the fun-damental period of buildings. The behaviour of both the soil and the structure is assumed to be elastic.The soil-foundation system is modelled using translational and rotational discrete springs. Analysis is firstconducted for one-storey buildings. It shows that the influence of the SSI on the fundamental frequencyof building depends on the soilstructure relative rigidity Kss. Analysis is then extended for multi-storeybuildings. It allows the generalization of the soilstructure relative rigidity Ks to such complex structures.Charts are proposed for taking into account the influence of the SSI in the calculation of the fundamentalfrequency of a wide range of buildings. Copyright q 2007 John Wiley & Sons, Ltd.

Received 10 October 2006; Revised 19 June 2007; Accepted 21 June 2007

KEY WORDS: building; charts; fundamental period; relative rigidity; soilstructure interaction; seismic

1. INTRODUCTION

Building codes generally use the fundamental period of buildings to assess their response to

seismic loadings. This parameter is generally calculated using empirical formulas provided by

seismic codes. These formulas generally ignore the soil flexibility, which could drastically affect

the fundamental period of buildings and consequently their overall seismic response. On the basis

of measurements collected from buildings during earthquakes, Goel and Chopra [1] concludedthat in the case of shear-wall structures, empirical formulas such as those proposed by the Uniform

Correspondence to: Marwan Sadek, Laboratoire de Mecanique de Lille (UMR 8107), Universite des Sciences etTechnologies de Lille, Villeneuve dAscq, F-59655, France.

E-mail: [email protected] Professor.Graduate Student. Professor.

Copyright q 2007 John Wiley & Sons, Ltd.


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2446 L. KHALIL, M. SADEK AND I. SHAHROUR

Building Code[2]are inadequate. Through a series of micro-vibrations tests, Ghrib and Mamedov[3] confirmed the inadequacy of the fundamental period formula given by the National BuildingCodes of Canada[4] or the Uniform Building Code[2], since they do not include the foundationflexibility.

The role of the soilstructure interaction (SSI) is usually considered beneficial to the structuralsystem under seismic loading since it lengthens the lateral fundamental period and leads to higher

damping of the system. This conclusion could be misleading. Indeed, recent case studies and post-

seismic observations suggest that the SSI can be detrimental and neglecting its influence could lead

to unsafe design for both the superstructure and the foundation especially for structures founded

on soft soil[5, 6]. Mylonakis and Gazetas[6]reported three cases of earthquakes (Bucharest 1977,Mexico City 1985 and Kobe 1995) where SSI caused an increase in the seismic-induced response

of structures despite a possible increase in damping. They reported that Mexico earthquake was

particularly destructive to l012 storey buildings founded on soft clay, whose period increased from

about 1.0 s (assumption of a fixed base structure) to nearly 2.0 s due to the SSI. Several authors

attempted to determine the influence of the SSI on the seismic response of buildings. First studies

were conducted by Veletsos and Meek[

7], Veletsos and Nair

[8]

and Bielak[

9]. The flexible-base

periodT is evaluated from Velestos and Meek[7] for a structure with a surface foundation asfollows:

TT

=

1+ kku

+ kh2

k(1)

where Tdenotes the fixed-base system period; ku and k stand for the rotational and translational

springs, h and kdesignate the height and flexural rigidity of the structure, respectively. Similar

formula is recommended by the Building Seismic Safety Council[10]. On the basis of data recordedon 57 building covering a wide range of structural and geotechnical conditions, Stewartet al.[11]concluded that approaches proposed by Velestos and Bielak to predict

T/T can reliably predict

the effect of inertial interaction but are limited to single degree of freedom (SDOF) oscillators. The

objective of the present paper concerns the elaboration of a simple procedure for taking into accountthe influence of the SSI in the determination of the fundamental frequency of buildings. Analyses

conducted for both one-storey and multi-storey buildings for various geotechnical conditions allow

the construction of charts that give the fundamental frequency of a wide range of buildings in

terms of the relative soilstructure stiffness.

2. ONE-STOREY BUILDING

2.1. Numerical model

Analysis of the influence of the SSI on the seismic response of buildings constitutes a

complex task, since it depends on multiple factors that affect the fundamental frequency ofbuildings. In order to analyse this problem, the case study that will be firstly investigated con-

cerns one-storey buildings modelled as a reinforced concrete frame founded on a homoge-

neous elastic soil layer (Youngs modulus = 50 MPa; density = 20kN/m3; Poissons ratio = 0.3;shear wave velocity = 98 m/s). This frame is composed of a slab supported by two reinforcedconcrete square columns (Youngs modulus = 32 000 MPa; density= 24.5 kN/m3) and based ontwo isolated square footings (2.0 m 2.0 m). The height of each column is 4.0m and its cross

Copyright q 2007 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2007; 36:24452453

DOI: 10.1002/eqe


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This equation shows that the fundamental frequency of the soil-foundationstructure system de-

pends on five parameters that characterize the soil (Vs), the foundation (A) and the structure

(H,Ic,Ec). Note that the influence of span length has been investigated prior to the present study

and was found insignificant. In the sequel, analyses are conducted for different values of influential

parameters in order to identity a dimensionless factor representing the soilstructure relative rigid-ity:Vs: 98, 107, 140, 200, 250 and 300 m/s (stiff soil); H: 2, 4, 7, 10 and 13 m;Ec: 20 000, 32 000and 42 000 MPa;Ic: 0.000139, 0.00135, 0.0131 and 0.1357 m4. Results of these analyses are sum-marized in Table I. They were approximated using the following multilinear regression:

log(f/fFB)=A1log(Vs)+ A2log(H)+ A3log(Ec)+ A4log(Ic) (4)fFB designates the fundamental frequency of the fixed-base structure (without SSI effect).

Table I. Influence of Kss on the fundamental frequency of one-storey buildings.

f/fFB Vs(m/s) H(m) Ec(MPa) Ic(m4)

1 98 4 32 000 0.0001391 107 4 32 000 0.0001391 140 4 32 000 0.0001391 200 4 32 000 0.0001391 250 4 32 000 0.0001391 300 4 32 000 0.0001390.973 98 4 32 000 0.001350.976 107 4 32 000 0.001350.985 140 4 32 000 0.001350.995 200 4 32 000 0.001350.998 250 4 32 000 0.001350.999 300 4 32 000 0.001350.819 98 4 32 000 0.013150.833 107 4 32 000 0.013150.891 140 4 32 000 0.013150.949 200 4 32 000 0.013150.967 250 4 32 000 0.013150.976 300 4 32 000 0.013150.576 98 4 32 000 0.1357520.589 107 4 32 000 0.1357520.686 140 4 32 000 0.1357520.823 200 4 32 000 0.1357520.876 250 4 32 000 0.1357520.908 300 4 32 000 0.1357520.548 98 2 32 000 0.013150.584 107 2 32 000 0.013150.682 140 2 32 000 0.013150.804 200 2 32 000 0.013150.862 250 2 32 000 0.01315

0.896 300 2 32 000 0.013150.819 98 4 32 000 0.013150.833 107 4 32 000 0.013150.891 140 4 32 000 0.013150.949 200 4 32 000 0.013150.967 250 4 32 000 0.013150.976 300 4 32 000 0.01315


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INFLUENCE OF THE SOILSTRUCTURE INTERACTION 2449

Table I. Continued.


0.963 98 7 32 000 0.01315

0.983 107 7 32 000 0.013150.963 140 7 32 000 0.013150.983 200 7 32 000 0.013150.989 250 7 32 000 0.013150.993 300 7 32 000 0.013150.966 98 10 32 000 0.013150.972 107 10 32 000 0.013150.983 140 10 32 000 0.013150.992 200 10 32 000 0.013150.995 250 10 32 000 0.013150.996 300 10 32 000 0.013150.979 98 13 32 000 0.013150.985 107 13 32 000 0.013150.987 140 13 32 000 0.01315

0.994 200 13 32 000 0.013150.997 250 13 32 000 0.013151 300 13 32 000 0.013150.981 98 4 20 000 0.001350.985 107 4 20 000 0.001350.991 140 4 20 000 0.001350.996 200 4 20 000 0.001350.997 250 4 20 000 0.001350.998 300 4 20 000 0.001350.961 98 4 42 000 0.001350.967 107 4 42 000 0.001350.981 140 4 42 000 0.001350.991 200 4 42 000 0.001350.994 250 4 42 000 0.00135

0.996 300 4 42 000 0.001350.881 98 4 20 000 0.013150.899 107 4 20 000 0.013150.936 140 4 20 000 0.013150.968 200 4 20 000 0.013150.979 250 4 20 000 0.013150.985 300 4 20 000 0.013150.787 98 4 42 000 0.013150.814 107 4 42 000 0.013150.877 140 4 42 000 0.013150.936 200 4 42 000 0.013150.958 250 4 42 000 0.013150.969 300 4 42 000 0.013150.666 98 4 20 000 0.1357520.699 107 4 20 000 0.1357520.785 140 4 20 000 0.1357520.878 200 4 20 000 0.1357520.917 250 4 20 000 0.1357520.939 300 4 20 000 0.1357520.523 98 4 42 000 0.1357520.558 107 4 42 000 0.1357520.657 140 4 42 000 0.135752


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Table I. Continued.


0.784 200 4 42 000 0.135752

0.845 250 4 42 000 0.1357520.883 300 4 42 000 0.1357520.316 98 2 32 000 0.13150.338 107 2 32 000 0.13150.413 140 2 32 000 0.13150.557 200 2 32 000 0.13150.661 250 2 32 000 0.13150.735 300 2 32 000 0.13150.842 98 7 32 000 0.13150.863 107 7 32 000 0.13150.911 140 7 32 000 0.13150.954 200 7 32 000 0.13150.971 250 7 32 000 0.13150.978 300 7 32 000 0.1315

0.911 98 10 32 000 0.13150.924 107 10 32 000 0.13150.953 140 10 32 000 0.13150.977 200 10 32 000 0.13150.986 250 10 32 000 0.13150.987 300 10 32 000 0.13150.953 98 13 32 000 0.13150.955 107 13 32 000 0.13150.974 140 13 32 000 0.13150.988 200 13 32 000 0.13150.993 250 13 32 000 0.13150.995 300 13 32 000 0.1315

Values of the coefficients of the multilinear regression are: A1= 0.233,A2= 0.33,A3= 0.12and A4= 0.07. They are obtained with a determination coefficient R2 = 0.7.Normalizing the coefficients of the regression by A3 gives A1/A3 2, A2/A3 3 and

A4/A3 0.75. As a consequence, the proposed dimensionless parameter Kss representing thesoilstructure relative rigidity could be approximated as follows:

Kss= V2sH3Ec (Ic)3/4

(5)

Expressions of discrete springs (Equation (2)) reveals that the influence of the square of soil shear

wave velocity(V2s ) is equivalent to the square root of foundation area (

A); hence, the expression

of the soilstructure relative rigidity Kss is turned into

Kss= V2sH3 A

A0

Ec (Ic)3/4 (6)

where A0 denotes a reference area (1 m2).

Figure 2(a) shows the variation of the ratio f/fFBwith the relative rigidityKss. It can be observed

that this variation can be approximated using a simple chart. For log(Kss)>1.5, the ratio f/fFB


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0

0.6

0.8

1

1.2

Log (Kss)

f/fFB

0.4

0.2

-4 -3 -2 -1 0 1 2 43

0

0.2

0.4

0.6

0.8

1

1.2

Log (Kss)

f/fFB

Ns

-4 -3 -2 -1 0 1 3 42

0

0.2

0.4

0.6

0.8

1

1.2

Log (Kss)

f/fF

B

Ns

NbtNbl

-4 -3 -2 -1 0 4321

(a)

(b) (c)

Figure 2. Influence of Kss on the fundamental frequency of buildings: (a) one-storeybuildings; (b) multi-storey buildings with one span in the lateral and transverse directions;

and (c) multi-storey buildings with multiple spans.

attains an asymptote f/fFB= 1. In this case, the fundamental frequency of the soil-foundationstructure system is close to that of a fixed-base structure. For log(Kss)


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0

0.2

0.4

0.6

1

1.2

-4 -3 -2 -1 0 1 2 4Log (Kss)

f/fFB

3

0.8

Figure 3. Chart for the consideration of the SSI in the determination of thefundamental frequency of buildings.

This expression indicates a decrease of Kss with the increase in the number of stories, and

consequently to an increase in the influence of the SSI on the fundamental frequency of the

building. Figure 2(a) depicts the variation of f/fFB with the relative rigidity Kss. It shows that this

variation could be approximated using a simple chart that is similar to that obtained for one-storey

buildings. Calculations were also conducted for multi-storey buildings with different values of the

number of spans in both the transversal and longitudinal directions (Nbt and Nbl) and different

foundation areas (A1,A2 and A3) proportional to the transmitted loads.

Analyses show that the relative rigidity for these buildings could be expressed as follows:

Kss=NbtNbl V2sH3

A

A0

NsEc (Ic)3/4 (8)

where A is the mean value of foundations area A = (A1+A2+ + An)/n.The increase in the number of spans (Nbl or Nbt) leads to higher values of the relative rigidity

Kss and consequently to a decrease in the influence of the SSI. Figure 2(c) depicts the variation of

the fundamental frequency ratio (f/fFB) with Kss for buildings with multiple stories and spans.

It confirms trends obtained for one-storey buildings. Figure 3 summarizes results obtained for

one-storey and multi-storey buildings. It can be observed that the fundamental frequency ratio

(f/fFB

) can be determined in terms of the relative rigidity Kss

(Equation (8)) using the chart

described earlier for one-storey buildings.

4. CONCLUSION

This paper included the analysis of the influence of the soilstructure interaction (SSI) on the

fundamental period of buildings. Analyses conducted for various soil and structure conditions


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showed that this influence depends mainly on the soilstructure relative rigidity, which could be

expressed in terms of the soil shear wave velocity (Vs), the foundation area (A), the flexural

rigidity of the building columns (Ic,Ec), the storey height (H), the number of stories (Ns) and

spans (Nbt,Nbl) as follows:

Kss=NbtNbl V2sH3

A

A0

NsEc (Ic)3/4

A chart is proposed for an ease consideration of the influence of the SSI in the determination of

the fundamental frequency of buildings. For flexible buildings (log(Kss)>1.5), the SSI could be

neglected. For lower values of Kss, ignoring the SSI could lead to a significant misestimation of

the fundamental frequency of buildings.

REFERENCES

1. Goel RK, Chopra AK. Period formulas for concrete shear wall buildings. Journal of Structural Engineering(ASCE) 1998; 124:426433. DOI: 10.1061/(ASCE)0733-9445(1998)124:4(426).

2. Uniform Building Code. International Conference of Building Officials, Whittier, CA, U.S.A., 1997.

3. Ghrib F, Mamedov H. Period formulas of shear wall buildings with flexible bases. Earthquake Engineering and

Structural Dynamics 2004; 33:295314. DOI: 10.1002/eqe.344.

4. NBCC (95)National Research Council Canada. Supplement to the National Building Code of Canada , Ottawa,

Canada, 1995.

5. Gazetas G, Mylonakis G. Seismic SoilStructure Interaction: New Evidence and Emerging Issues. Emerging

Issues Paper, Geotechnical Special Publication No. 75, vol. 3. ASCE: New York, 1998; 11191174.

6. Mylonakis G, Gazetas G. Seismic soilstructure interaction: beneficial or detrimental? Journal of Earthquake

Engineering2000; 4(3):277301. DOI: 10.1142/S1363246900000175.

7. Veletsos AS, Meek JW. Dynamic behavior of building-foundation systems. Earthquake Engineering and Structural

Dynamics 1974; 3(2):121138.

8. Veletsos AS, Nair VV. Seismic interaction of structures on hysteretic foundations.Journal of Structural Engineering

(ASCE) 1975; 101(1):109129.

9. Bielak J. Dynamic behavior of structures with embedded foundations. Earthquake Engineering and Structural

Dynamics 1975; 3(3):259274.

10. BSSCBuilding Seismic Safety Council. The 2003 NEHRP Recommended Provisions for New Buildings and

Other Structures Part 2: Commentary (FEMA 450). Federal Emergency Management Agency, Washington, DC,

2003.

11. Stewart JP, Seed RB, end Fenves GL. Seismic soilstructure interaction in buildings II: empirical findings. Journal

of Geotechnical and Geoenvironmental Engineering (ASCE) 1999; 125(1):3848. DOI: 10.1061/(ASCE)1090-

0241(1999)125:1(26).

12. Newmark NM, Rosenblueth E. Fundamentals of Earthquake Engineering. Prentice-Hall: Englewood Cliffs, NJ,

1971.


DOI: 10.1002/eqe

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